gh-35977: Move hilbert_numerator test to its proper place
    
And make it clear that it gives a wrong answer
    
URL: #35977
Reported by: Antonio Rojas
Reviewer(s):

Author: Release Manager

src/sage/rings/polynomial/hilbert.pyx

@@ -576,14 +576,6 @@ def hilbert_poincare_series(I, grading=None):
         120*t^3 + 135*t^2 + 30*t + 1
         sage: hilbert_poincare_series(J).denominator().factor()
         (t - 1)^14
-
-    This example exceeded the capabilities of Singular before version 4.2.1p2.
-    In Singular 4.3.1, it works correctly on 64-bit, but on 32-bit, it prints overflow warnings
-    and omits some terms::
-
-        sage: J.hilbert_numerator(algorithm='singular')
-        120*t^33 - 3465*t^32 + 48180*t^31 - 429374*t^30 + 2753520*t^29 - 13522410*t^28 + 52832780*t^27 - 168384150*t^26 + 445188744*t^25 - 987193350*t^24 + 1847488500*t^23 + 1372406746*t^22 - 403422496*t^21 - 8403314*t^20 - 471656596*t^19 + 1806623746*t^18 + 752776200*t^17 + 752776200*t^16 - 1580830020*t^15 + 1673936550*t^14 - 1294246800*t^13 + 786893250*t^12 - 382391100*t^11 + 146679390*t^10 - 42299400*t^9 + 7837830*t^8 - 172260*t^7 - 468930*t^6 + 183744*t^5 - 39270*t^4 + 5060*t^3 - 330*t^2 + 1  # 64-bit
-        ...120*t^33 - 3465*t^32 + 48180*t^31 - ...  # 32-bit
     """
     cdef Polynomial_integer_dense_flint HP
     HP, grading = first_hilbert_series(I, grading=grading, return_grading=True)

src/sage/rings/polynomial/multi_polynomial_ideal.py

@@ -3123,6 +3123,14 @@ def hilbert_numerator(self, grading=None, algorithm='sage'):
             sage: I.hilbert_numerator()                                                 # needs sage.rings.number_field
             -t^5 + 1
 
+        This example returns a wrong answer due to an integer overflow in Singular::
+
+            sage: n=4; m=11; P = PolynomialRing(QQ, n*m, "x"); x = P.gens(); M = Matrix(n, x)
+            sage: I = P.ideal(M.minors(2))
+            sage: J = P * [m.lm() for m in I.groebner_basis()]
+            sage: J.hilbert_numerator(algorithm='singular')
+            ...120*t^33 - 3465*t^32 + 48180*t^31 - ...
+
         Our two algorithms should always agree; not tested until
         :trac:`33178` is fixed::